Solution
= Solution
\b[(ii) implies (i):] together with the preliminary estimate, (ii) gives <equivalent norms>,
$$
\|x\|\leq\|x\|'\leq C\|x\|.
$$
An equivalent <norm> on a <Banach space> is complete: a primed-<norm> <Cauchy sequence> has an original-<norm> limit, and the upper bound makes convergence hold also in the primed <norm>. \b[(ii) implies (iii)] immediately because $\|P_k\|\leq C$ for every $k$.